Climate modelling predicts a slowdown in this ocean current
In 2001 climate models predicted a decrease in the overturning circulation as carbon dioxide in the atmosphere increased. This could happen over a decade or so, perhaps even quicker.
This is a "hypothesis" (a prediction of what we think will happen based on our theory of how it works). It is tested against actual evidence. If the evidence agrees with the hypothesis this indicates that the theory is right. Or does it?
At the time of the predictions scientists knew very little about the currents. They worked out how to monitor the circulation across the Atlantic using instruments anchored to the sea floor. Later, new satellite instruments added additional data about water density, temperature and height.
In 2004 an 
analysis of satalite data seemed to show a slowing in a North Atlantic section of the currents. In 2005, other scientists reported that the number of giant chimneys of cold, descending water under the Arctic which help power the currents had declined from the normal 7-12 down to 2 weak columns. 
Another study also in 2005 found a 30% reduction in warm water flow in the Gulf Stream. These data all supported the idea that the currents are slowing down.
Careful language, the case was not yet proved
In 2005, the NOC scientists wrote:
The comparison (of the new set of data with the earlier sets) suggests that the overturning circulation has slowed by about 30 per cent between 1957 and 2004.
Nature 1/12/05
Notice that they wrote 'suggests' and not 'shows'. The IoP ethical code says scientists have a duty, under the heading of social responsibility, to 'alert the wider public to the consequences of their work'. This includes communicating the idea of 'uncertainty'. The NOC scientists have done this. Harry Bryden, the leader of the group told New Scientist:
We don't want to say the circulation will shut down but we are nervous about our findings. They have come as quite a surprise.
What happened next?
In 2008, a different group of scientists found that the currents in the deep water had come back!
What was going on?
In 2010, NASA scientists were able to 
put together both the water flow data AND satelitte data from the previous studies. This combined data showed no overall slowdown in the currents over the previous 15 years and even that it was speeding up.
What all of this shows is that any scientific result is always provisional and that it can be updated at any time if new data or analysis techniques become available.
These studies do not "prove" that the overturning circulation will not shut down due to climate change. They just show that fortunately it hasn't happened so far.
Methane Release
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